A two form $\alpha$ on a n- manifold $M$ is called anti symplectic if for every $x\in M$, $\{ v\in T_{x} M \mid i_{v} \alpha=0 \}$ is a $n-2$ dimensional subspace of $T_{x}M$. So we obtain a $n-2$ dimensional subbundle of $T M$. We denote it by $\tilde{\ker} \;\alpha$

*The first question:*

Under what condition on $\alpha$, the distribution $\tilde{\ker} \;\alpha$ is integrable? (I am motivated by Frobenious condition $\alpha \wedge d\alpha=0$, so I search for an algebraic condition)

*The second question:*

Assume that $\omega$ is a symplectic 2-form on a 2n- manifold. Can we write $\omega$ in the global form $\omega=\sum_{i=1}^{n} \alpha_{i}$ where each $\alpha_{i}$ is anti symplectic form.(I think that the local argument and then using partition of unity, does not work)

half-rankof $\alpha$. Your 'anti-symplectic' $2$-forms are simply the $2$-forms of constant half-rank $1$. Also, they are the $2$-forms that arelocally decomposablein the sense that they can locally be written as $\alpha = \omega_1\wedge\omega_2$ for some linearly independent $1$-forms $\omega_1$ and $\omega_2$. – Robert Bryant Apr 1 at 13:59